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Theorem resf2nd 16536
Description: Value of the functor restriction operator on morphisms. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
resf1st.f (𝜑𝐹𝑉)
resf1st.h (𝜑𝐻𝑊)
resf1st.s (𝜑𝐻 Fn (𝑆 × 𝑆))
resf2nd.x (𝜑𝑋𝑆)
resf2nd.y (𝜑𝑌𝑆)
Assertion
Ref Expression
resf2nd (𝜑 → (𝑋(2nd ‘(𝐹f 𝐻))𝑌) = ((𝑋(2nd𝐹)𝑌) ↾ (𝑋𝐻𝑌)))

Proof of Theorem resf2nd
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-ov 6638 . 2 (𝑋(2nd ‘(𝐹f 𝐻))𝑌) = ((2nd ‘(𝐹f 𝐻))‘⟨𝑋, 𝑌⟩)
2 resf1st.f . . . . . 6 (𝜑𝐹𝑉)
3 resf1st.h . . . . . 6 (𝜑𝐻𝑊)
42, 3resfval 16533 . . . . 5 (𝜑 → (𝐹f 𝐻) = ⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩)
54fveq2d 6182 . . . 4 (𝜑 → (2nd ‘(𝐹f 𝐻)) = (2nd ‘⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩))
6 fvex 6188 . . . . . 6 (1st𝐹) ∈ V
76resex 5431 . . . . 5 ((1st𝐹) ↾ dom dom 𝐻) ∈ V
8 dmexg 7082 . . . . . 6 (𝐻𝑊 → dom 𝐻 ∈ V)
9 mptexg 6469 . . . . . 6 (dom 𝐻 ∈ V → (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))) ∈ V)
103, 8, 93syl 18 . . . . 5 (𝜑 → (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))) ∈ V)
11 op2ndg 7166 . . . . 5 ((((1st𝐹) ↾ dom dom 𝐻) ∈ V ∧ (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))) ∈ V) → (2nd ‘⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩) = (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))))
127, 10, 11sylancr 694 . . . 4 (𝜑 → (2nd ‘⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩) = (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))))
135, 12eqtrd 2654 . . 3 (𝜑 → (2nd ‘(𝐹f 𝐻)) = (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))))
14 simpr 477 . . . . . 6 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → 𝑧 = ⟨𝑋, 𝑌⟩)
1514fveq2d 6182 . . . . 5 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → ((2nd𝐹)‘𝑧) = ((2nd𝐹)‘⟨𝑋, 𝑌⟩))
16 df-ov 6638 . . . . 5 (𝑋(2nd𝐹)𝑌) = ((2nd𝐹)‘⟨𝑋, 𝑌⟩)
1715, 16syl6eqr 2672 . . . 4 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → ((2nd𝐹)‘𝑧) = (𝑋(2nd𝐹)𝑌))
1814fveq2d 6182 . . . . 5 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻𝑧) = (𝐻‘⟨𝑋, 𝑌⟩))
19 df-ov 6638 . . . . 5 (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
2018, 19syl6eqr 2672 . . . 4 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻𝑧) = (𝑋𝐻𝑌))
2117, 20reseq12d 5386 . . 3 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)) = ((𝑋(2nd𝐹)𝑌) ↾ (𝑋𝐻𝑌)))
22 resf2nd.x . . . . 5 (𝜑𝑋𝑆)
23 resf2nd.y . . . . 5 (𝜑𝑌𝑆)
24 opelxpi 5138 . . . . 5 ((𝑋𝑆𝑌𝑆) → ⟨𝑋, 𝑌⟩ ∈ (𝑆 × 𝑆))
2522, 23, 24syl2anc 692 . . . 4 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝑆 × 𝑆))
26 resf1st.s . . . . 5 (𝜑𝐻 Fn (𝑆 × 𝑆))
27 fndm 5978 . . . . 5 (𝐻 Fn (𝑆 × 𝑆) → dom 𝐻 = (𝑆 × 𝑆))
2826, 27syl 17 . . . 4 (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
2925, 28eleqtrrd 2702 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom 𝐻)
30 ovex 6663 . . . . 5 (𝑋(2nd𝐹)𝑌) ∈ V
3130resex 5431 . . . 4 ((𝑋(2nd𝐹)𝑌) ↾ (𝑋𝐻𝑌)) ∈ V
3231a1i 11 . . 3 (𝜑 → ((𝑋(2nd𝐹)𝑌) ↾ (𝑋𝐻𝑌)) ∈ V)
3313, 21, 29, 32fvmptd 6275 . 2 (𝜑 → ((2nd ‘(𝐹f 𝐻))‘⟨𝑋, 𝑌⟩) = ((𝑋(2nd𝐹)𝑌) ↾ (𝑋𝐻𝑌)))
341, 33syl5eq 2666 1 (𝜑 → (𝑋(2nd ‘(𝐹f 𝐻))𝑌) = ((𝑋(2nd𝐹)𝑌) ↾ (𝑋𝐻𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  Vcvv 3195  cop 4174  cmpt 4720   × cxp 5102  dom cdm 5104  cres 5106   Fn wfn 5871  cfv 5876  (class class class)co 6635  1st c1st 7151  2nd c2nd 7152  f cresf 16498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-2nd 7154  df-resf 16502
This theorem is referenced by:  funcres  16537
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